Apply: K-2 Sample Task
GRADE 2 MATH: CAROL’S NUMBERS
UNIT OVERVIEW
The mathematics of the unit involves understanding the meaning of base ten and using that understanding to
solve number and real life problems. The number line is used as a tool to help articulate understanding of
base ten and to solve problems using addition and subtraction of numbers less than one hundred. The big
idea is for students to understand and use groups of ten. Strategies will involve applying number properties
including distributive, associative, and commutative.
TASK DETAILS
Task Name: Carol’s Numbers
Grade: 2
Subject: Math
Task Description: The final performance assessment is entitled Carol’s Numbers. The mathematics of the task
involves understanding the meaning of base ten and using that understanding to compare the magnitude of
numbers. The number line is used as a tool to help articulate understanding of base ten.
Standards Assessed:
2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and
ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones.
2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
2.NBT.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >,
=, and < symbols to record the results of comparisons.
2.MD.6 Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points
corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a
number line diagram.
Standards for Mathematical Practice:
MP.1 Make sense of problems and persevere in solving them.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.6 Attend to precision.
1
COMMON CORE CURRICULUM EMBEDDED TASK
4
COMMON CORE CURRICULUM EMBEDDED TASK
5
COMMON CORE CURRICULUM EMBEDDED TASK
Student B – Level 4 (Score 8)
The student’s
explanation of how to
construct the smallest
number indicates
strong knowledge of
place value.
Articulating the
process of
determining the
smallest digit
available and then
putting in the
hundred’s place is
sophisticated thinking
at this grade.
2.NBT.1&4 MP3
14
COMMON CORE CURRICULUM EMBEDDED TASK
Student B – Level 4 (Score 8) Page 2
The student
accurately
places the
numbers on
the number
line and then
explains that
31 is almost
(in the middle)
between 21
and 42,
indicating
knowledge of
quantities.
2NBT.4,
2MD.6, MP2
15
COMMON CORE CURRICULUM EMBEDDED TASK
Level 3: Performance at Standard (Score Range 5 - 6)
For most of the task, the student’s response shows the main elements of performance that the tasks
demand and is organized as a coherent attack on the core of the problem. There are errors or omissions,
some of which may be important, but of a kind that the student could well fix, with more time for
checking and revision and some limited help. The student explains the problem and identifies
constraints. The student makes sense of quantities and their relationships in the problem situations.
S/he often uses abstractions to represent a problem symbolically or with other mathematical
representations. The student response may use assumptions, definitions, and previously established
results in constructing arguments. They may make conjectures and build a logical progression of
statements to explore the truth of their conjectures. The student might discern patterns or structures
and make connections between representations.
Student C – Level 3 (Score 6)
The student
correctly creates the
largest and smallest
numbers possible
out of the three
cards. The
explanation is short
but does
communicate least
to greatest,
indicating
understanding and
good use of
vocabulary. 2NBT.4
16
COMMON CORE CURRICULUM EMBEDDED TASK
Student C – Level 3 (Score 6) Page 2
The student’s
location of 21
indcates fragile
understanding of
numbers on the
number line. It
should be located
half way between 0
and 42. The student
does show some
understanding by
correctly locating 31.
The student explains
that 31 is between
21 and 42. The
student reasons
quantitatively and
provides a clear
explanation of how
they thought.
MP2 and MP3
17
COMMON CORE CURRICULUM EMBEDDED TASK
Student D – Level 3 (Score 5)
The digits are
correctly
placed to make
the largest and
smallest
possible
numbers. The
explanation
addresses the
smallest and
largest digits,
leaving the
third digit to be
placed by
default. The
student implies
place value and
comparison.
2NBT.1 & 4
18
COMMON CORE CURRICULUM EMBEDDED TASK
Student D – Level 3 (Score 5) Page 2
The student
misplaced 21,
but did place 31
about halfway
between 21 and
42. The
student’s
explanation
indicated all the
reasoning was
concentrated on
placing 31
correctly. 2MD.6
MP2
19
COMMON CORE CURRICULUM EMBEDDED TASK
Student E – Level 3 (Score 5)
The student
found the
largest and
smallest 3
digit
numbers.
The
explanation
merely
recalls the
process the
student used
to make the
smallest
number, but
does not
explain why.
The student
needs to by
indicate ideas
such as least
or most. The
students
needs to
show more
quantitative
reasoning.
MP2
20
COMMON CORE CURRICULUM EMBEDDED TASK
Student E – Level 3 (Score 5) Page 2
21 is misplaced. The student does explain why s/he placed
31 correctly on the number line. The student needs
additional experiences in finding the mid-point between a
larger number and zero. 2MD.6
21
COMMON CORE CURRICULUM EMBEDDED TASK
Level 3 Implications for Instruction
Students who met standard on the task can still improve their performance by being attentive to
precession and by making complete explanations. Students must learn to provide complete
explanations of their process and why it makes sense. Many of the students who met level 3 failed to
place 21 correctly on the number line. They seemed to fail to understand that 21 should be half way
between 0 and 42. On the other hand, most students could accurately use their placement of 21 to find
a mid-way mark between its placement and 42 to locate 31. This may indicate that students have
quantative reasoning about length when comparing smaller lengths (20) apart. These same students
had trouble with longer distances (42) and perhaps dealing with 0 and an ending length.
Therefore, students need more practice with finding lengths on the number line. Student should locate
numbers on an open number line, learning to make relative judgements base on benchmark amounts.
Students will benefit from engaging in number line math talks and measuring along a number line when
working with numbers.
22
COMMON CORE CURRICULUM EMBEDDED TASK
Level 2: Performance below Standard (Score Range 2 - 4)
The student’s response shows some of the elements of performance that the tasks demand and some
signs of a coherent attack on the core of some of the problems. However, the shortcomings are
substantial, and the evidence suggests that the student would not be able to produce high-quality
solutions without significant further instruction. The student might ignore or fail to address some of the
constraints. The student may occasionally make sense of quantities in relationships in the problem, but
their use of quantity is limited or not fully developed. The student response may not state assumptions,
definitions, and previously established results. While the student makes an attack on the problem it is
incomplete. The student may recognize some patterns or structures, but has trouble generalizing or
using them to solve the problem.
Student F – Level 2 (Score 4)
The student’s
explanation
restates what was
shown in part 4
and 5. It does not
explain how to
create the
smallest number
or how the
student knows it is
the smallest
number. The
student needs
more instruction
around creating
clear and
complete
mathematical
explanations. MP3
23
COMMON CORE CURRICULUM EMBEDDED TASK
Student F – Level 2 (Score 4) Page 2
This work continues to show the trend of not being
able to accurately place 21 half way between 0 and 42,
but still showing that 31 is about mid-way between 21
and 42. More quantitative reasoning is needed. 2MD.6,
MP2
24
COMMON CORE CURRICULUM EMBEDDED TASK
Level 2 Implications for Instruction
Students need help in comparing the placement of numbers using 0 as a benchmark. The explanations
at this level are either incomplete or not focused on mathematical reasoning the make sense for the
situation. The students need learning experiences with number lines that start with zero.
Students can experiment with these ideas to develop a deeper conception of numbers, so that they may
more flexibly reason quantitatively. Instruction should involve more work on defining the relationships
between length and the size of numbers. Students need experiences to construct learning for
themselves. Students should be asked to explain and justify their answers regularly in class to develop
mathematical argumentation. Examining and analyzing other students’ explanations is an important
experience for students. It provides models and helps students discern important elements of
convincing arguments.
25
COMMON CORE CURRICULUM EMBEDDED TASK
Level 1: Demonstrates Minimal Success (Score Range 0 – 1)
The student’s response shows few of the elements of performance that the tasks demand. The work
shows a minimal attempt on the problem and struggles to make a coherent attack on the problem.
Communication is limited and shows minimal reasoning. The student’s response rarely uses definitions
in their explanations. The student struggles to recognize patterns or the structure of the problem
situation.
Student G – Level 1 (Score 1)
The student must
have added the
original digits to find
13 and then used it to
indicate the largest
digit. The student
was able to determine
the smallest (three
digit) number, but
goes on to explain
that 1 is the smallest
possible number. The
student’s explanation
showed a lack of
understanding what
the problem entailed.
The students needs to
focus on what the
prompts are asking.
26
COMMON CORE CURRICULUM EMBEDDED TASK
Student G – Level 1 (Score 1) Page 2
The student
showed little
understanding of
the prompt and
seem to ignore
the values 21, 31
and 85. The
student placed
random numbers
on the number
line. They did
seem to do so in
an accurate
manner though.
The explanation
was off the mark.
This student
needs additional
instruction in
understanding
the task prompts
and addressing
them.
27
COMMON CORE CURRICULUM EMBEDDED TASK
Level 1 Implications for Instruction
Students need support in reasoning quantitatively. They may need to start with number lines and the
idea of length equaling the size of a number. Students need experiences connecting numbers to their
location on the number line. Having students divide the distance between zero and the number to find
the location is important. The location of the number on the line is the equal to the number of
partitioned segments of length 1. Students need experiences in reasoning quantitatively about
benchmark numbers and their relate size to other numbers. One method successful students use is to
determine midway points as benchmark. They estimate about half to compare relative magnitude of
different lengths. Students need learning experiences with number lines, including both closed and
open number lines.
Students need additional instruction and experiences in writing explanations that fully articulate their
understanding and justify their findings. Sharing models of good explanations are helpful. Having
students rewrite and revise explanations is essential. Having students read others’ explanations and
critiquing their reasoning raises the cognitive demand and helps students create sounder arguments.
28
Task Analysis Tool Understanding Language Initiative, Stanford University Grade Level: 2nd Step 1: Examine and Identify Appropriate Instructional Task Name of Task: Carol’s Numbers Task Analysis Guiding Questions and Resources Step Guiding Questions: Is/does this task: o Clear in its expectations? o Grade-­‐level appropriate? o Aligned to the standards? o Require students to use language and analytical skills to demonstrate their content knowledge? Subject: Math Analysis Is this an appropriate task for analysis? Why? Yes. Students are asked to apply their content knowledge while also explaining/justifying their answer in writing. 1 Step 2: Identify Task Demands Guiding Questions: What do students need to do and know in terms of…? • Write down everything that students Content Knowledge Analytical Skills Language need to demonstrate, know, or do in Understand base ten Identify Explain in writing order to successfully complete this number system Name Read and write task. Understand hundreds, Recognize numbers • To do this, read (or watch) the task tens, ones Determine smallest Justify instructions. Correctly place a number Describe number on a number Compare Use content specific Resources: line Distinguish vocabulary: • For Content Knowledge: Common Create the smallest Apply concepts -­‐smallest Core State Standards, Next number using three Explain -­‐hundreds, tens, ones Generation Science Standards, or given digits (numbers) Prove -­‐number line other relevant standards (e.g., Show district, state, etc.) CCSS: Modify (while Use correct sentence • For Analytical Skills: Depth of 2.NBT.1 Understand performing task) structure: Knowledge (DOK) Levels (Find in that three digits of a Justify -­‐The number ___ goes Resources) three-­‐digit number here on the number line • For Language: Language Functions represents amounts of because… and Forms PDF (Find in Resources) hundreds, tens, and -­‐This is the smallest ones. number possible 2.NBT.3 Read and write because… numbers to 1000 using base ten numerals, number names, and expanded form. 2.NBT.4 Compare 2 three digit numbers based on meanings of the hundreds, tens, and ones digits. 2.NBT.6 Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2… 2 Step 3: Identify Disciplinary Practice(s) Step 4: Identify ELP Standard(s) Guiding Question: What disciplinary practice(s) are most relevant to this task? Resources: • Core Disciplinary Practices PDF (Find in Resources) • Interactive Correspondence between Practices, Tasks, and Functions PDF (Find in Resources) What are the relevant disciplinary analytical practices for this task: Math Practice 1: Make sense of problems and persevere in solving them. Math Practice 3: Construct viable arguments and critique the reasoning of others Math Practice 5: Use appropriate tools strategically. Math Practice 6: Attend to precision. Guiding Questions: What English Language Proficiency Standards are reflected in this task? Which of these ELP standards do you feel comfortable measuring or intend to assess? Resources: • The ELP Standards • Alternative Organization of Standards • The K-­‐12 Practices Matrix (Find in Resources) What are the relevant ELP standards for this task? Please include your reasoning behind selecting these standards. ELP Standard Your Reasoning ELP 1 The student must construct meaning from the math problem, i.e. understand what the questions is asking him or her to do; corresponds to Math Practice 1 ELP 4 The nature of the task is to state a claim of one’s mathematical reasoning and explain it; corresponds to Math Practice 3 ELP 9 The student’s explanation must correctly use language such as sequencing words and/or linking words (i.e. “because”) to explain reasoning; corresponds to Math Practice 6 and perhaps Math Practice 3. ELP 1-­‐ Construct meaning from oral presentations and literary and informational text through grade-­‐appropriate listening, reading, and viewing: The student must understand what the question is asking in order to proceed with the steps. ELP 4 – Construct grade-­‐appropriate oral and written claims and support them with reasoning and evidence: The student must not only give a mathematical response but also provide reasoning and explanation for that answer. Therefore the student must include a claim and reasoning in his/her final answer. 3 Using ELP Standards Level Descriptors (PLDs)
to Interpret Student Work
Understanding Language/SCALE, Stanford University
October 2016
Task: Carol’s Numbers
Grade Level: 2
Step 1
Examine the Identified ELP Standard(s) and Corresponding Level
Descriptors
Consulting ELP
Standards and
Level Descriptors
 Examine the identified ELP Standard(s) and corresponding level descriptors for the
task
 If there are many applicable standards, choose one or two that relate to your
students’ areas of growth.
Notes: Standards 1, 4, and 9 were identified as most relevant to this task.
Standard 1 - students need to construct meaning from the math problem.
Standard 4 - students need to state a claim about mathematical reasoning and support
it
Standard 9 – student’s response must use sequencing and linking words
Looking at standard 4 for this work as it is a function of language.
Step 2
Interpret Student Work Using the Standards Level Descriptors
Using ELP
Standards Level
Descriptors as
rubrics to
interpret student
work
 Do the same standards apply when you examine your students’ output?
 What level(s) most accurately describe your students’ work?
 Remember that the interpretation only tells you the level of this specific piece of
student work; your students’ levels might shift based on different tasks or learning
objectives.
 Identify patterns (similarities or differences) in your students’ work if you are
interpreting multiple pieces.
Notes: Looking at standard 4 on student sample B, on the explain questions, the
student explains how to order the numbers. The student gives an opinion and explains
why with several reasons explaining each place value. On the next page, the student
gives a reason for where he/she placed the number 31. Since this may not be a
familiar topic, I placed the student at a PLD of 4 on standard 4.
In reviewing this student work, it almost seems that standard 2 would be a good fit as
well- exchanging information and ideas - when looking at standard 2, the student is
able to participate in a short written exchange one again about a variety of topics. I
would also place the student at a PLD of 4 on standard 2.
Step 3
Identify strategies to support student needs
Identifying
instructional
supports to
 Use the identified level (and perhaps the next level) to provide student with
formative feedback.
 Use the identified patterns in student work to plan for instructional adjustments.
Understanding Language/SCALE, Stanford University
improve student
learning
 Consult relevant state/district resources for suggested strategies .
Notes: Some strategies and resources I might use to support this student's language
development would include Jeff Zwiers Math Constructive Conversation Skills poster
focusing on the Explain and Support, and Multiple Methods for Solving boxes. I might
also include a word bank focusing on sequence words to help with the first explain
question.
Understanding Language/SCALE, Stanford University